1120 THE BELL SYSTEM TECHNICAL JOURNAL, NOVEMBER 1952 



We may express 7 in terms of 1\ by equation (280), and so if Zn{y) 

 varies with 7 in any reasonably simple way, or better yet if Zn(y) is 

 essentially independent of 7 in the range of interest, equation (284) 

 may be solved numerically for F^ by successive approximations. 



A numerical solution of equation (284) is, however, rarely necessary, 

 since the right-hand side of the equation is just the ratio of the sheath 

 impedance Zn{y) to the resistance "per square", namely l/(^ga), of 

 all the conducting layers in a stack of thickness |a in parallel, and this 

 ratio will almost always be large compared to imity. This is another way 

 of saying that the total one-way conduction current in the stack is 

 large compared to the sum of the conduction and displacement currents 

 in either sheath. Even if the sheaths are infinitely thick metal plates of 

 conductivity gi , we have from equation (79) of Section III, since 

 7 W ioiy/jie , 



hgaZniy) = idgxam = (1 + i)ea/28i , (285) 



and for most frequencies of interest the thickness ^da of conducting ma- 

 terial in half the stack will be several times the skin thickness 81 in 

 the metal. If the medium outside the stack is free space, then Zn(y) 

 will be a few hundred ohms and a fortiori the right side of (284) will be 

 large compared to unity. So long as the inequality 



1^ 

 2 



gaZ„(y) I » 1 (286) 



is satisfied, the lowest root of (284) will be approximately 



r, = iw/a; (287) 



and so from (282) and (283) the attenuation and phase constants of a 

 plane Clogston 2 line with infinitesimally thin laminae and high-im- 

 pedance walls are 



. (288) 



2\^fx/e ga ' 

 ^ = wV^e. (289) 



To this approximation, there is neither amplitude nor phase distortion. 

 The principal mode in a coaxial Clogston 2 corresponds to the lowest 

 root in T( of equation (279). To solve this equation numerically with 

 finite boundary impedances Za(y) and Zb(y), w^hile possible in principle, 

 would evidently be a major undertaking. We shall therefore assume 

 throughout the present paper that the total conduction and displace- 

 ment currents flowing in the core and the sheath are negligible compared 



